Advertisement

Weak and Strong Convergence Theorems of Quasi-Nonexpansive Mappings in a Hilbert Spaces

  • Gang Eun Kim
Open Access
Article

Abstract

In this paper, we first prove the weak convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Then we prove the weak convergence for the Moudafi’s iterative scheme of quasi-nonexpansive mapping and nonexpansive mapping. Finally, we prove the strong convergence for the Moudafi’s iterative scheme of two quasi-nonexpansive mappings. Our results generalize the recent results due to Iemoto and Takahashi.

Keywords

Weak and strong convergence Fixed point Moudafi iteration process Quasi-nonexpansive mapping 

References

  1. 1.
    Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953) MATHCrossRefGoogle Scholar
  2. 2.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa Iteration process. J. Math. Anal. Appl. 178, 301–308 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Takahashi, W., Kim, G.E.: Approximating fixed points of nonexpansive mappings in Banach spaces. Math. Jpn. 48, 1–9 (1998) MathSciNetMATHGoogle Scholar
  6. 6.
    Iemoto, S., Takahashi, W.: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71(12), 2082–2089 (2009) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed point problems. Inverse Probl. 23, 1635–1640 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2005) Google Scholar
  9. 9.
    Browder, F.E.: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 74, 660–665 (1968) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Khan, S.H., Fukhar-ud-din, H.: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Nonlinear Anal. 61, 1295–1301 (2005) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 44, 375–380 (1974) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Groetsch, C.W.: A note on segmenting Mann iterates. J. Math. Anal. Appl. 40, 369–372 (1972) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Takahashi, W.: Nonlinear Functional Analysis. Kindaikagaku, Tokyo (1988) (Japanese) Google Scholar
  15. 15.
    Kim, G.E.: Convergence theorems for quasi-nonexpansive mappings in Banach spaces (submitted) Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Applied MathematicsPukyong National UniversityBusanKorea

Personalised recommendations